1 / 52

Exponential Functions

Exponential Functions. The “I’m going to lie to you a bit” version. Exponential functions measure steady growth. If you really want to know what that means exactly, take differential equations ( a fter Calculus) Here’s the basic (lying) version

michel
Download Presentation

Exponential Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Exponential Functions The “I’m going to lie to you a bit” version

  2. Exponential functions measure steady growth • If you really want to know what that means exactly, take differential equations (after Calculus) • Here’s the basic (lying) version • An exponential growth happens when something is making more of itself (in a “steady” way) • People, money, bacteria, etc…

  3. Example • One dollar makes one dollar every year. $1 $1 Year 0 Year 1

  4. Example • One dollar makes one dollar every year. $1 $1 $1 Year 0 Year 1

  5. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 Year 0 Year 1 Year 2

  6. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2

  7. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  8. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  9. Example • Every year I keep what I have and add what I have. $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  10. Example • Every year I double my money $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  11. My dollars over time

  12. Example • Every year I double my money $1 y=1(2x) y=# of $ x=# of yrs $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  13. Approximation function ƒ(x)=2x

  14. Standard form(s) of an exponential • y=abx • a is the initial value (y-intercept) • b is the growth factor • y=aerx • a is the initial value (y-intercept) • r is the “per-capita rate” also called “exponential growth rate” • Conversion • b=er, r=ln(b)

  15. When to use an exponential model • Things making more things • People making more people, bacteria making more bacteria, money making more money (interest) • Lots of identical events happening at random times • Radioactive decay (atoms decay at random times), Heat transfer (atoms bump into each other randomly)

  16. Asymptotes I have to lie to you until you take Calculus edition

  17. Asymptote • An asymptote is a line that an equation gets close to but never reaches. • Every exponential y=abx has a horizontal asymptote at y=0.

  18. Growth and Decay I’m still lying to you a bit

  19. Simple Version • A growth function is an exponential function where y gets bigger when x gets bigger. • A decay function is an exponential function where y gets smaller when x gets bigger. • The easy way to tell is to graph it!

  20. Growth and Decay Exponential Growth Exponential Decay

  21. Non-calculator examples • y=3(4)x. • When x=0, y=3. When x=1, y=12. • When x gets bigger, y gets bigger. • Exponential Growth • y=2(1/3)x • When x=0, y=2. When x=1, y=2/3. • When x gets bigger, y gets smaller. • Exponential Decay

  22. Non-calculator examples • Y=7(2)-x. • When x=0, y=7. When x=1, y=7(2)-1=7/2. • When x gets bigger, y gets smaller. • Exponential Decay • Y=1(1/2)-x • When x=0, y=1. When x=1, y=(1/2)-1=2. • When x gets bigger, y gets bigger. • Exponential Growth

  23. 2) y=3-x decay. 1) y=3x growth. 4) y=(1/2)-x growth. 3) y=(1/2)x decay. d) 2&3 represent decay

  24. Logarithms Seriously, take calculus, please

  25. Example • Every year I double my money $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  26. Example • If I know what time it is and want to know how much money I have $1 a=1(2t) a=# of $ t=# of yrs $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  27. Example • What if I know money and want to know time? $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  28. Example • How long would it take me to get $1,000,000? $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  29. Flashback! I start 10 miles away from my house and drive away from my house at 30 mph. If I know how far I am from my house, how long have I been driving? d=number of miles away from my house t=number of hours I’ve been driving d=30t+10 (d-10)/30=t

  30. Flashback! I start 10 miles away from my house and drive away from my house at 30 mph. If I know how far I am from my house, how long have I been driving? d=number of miles away from my house t=number of hours I’ve been driving d=30t+10 (d-10)/30=t I need an inverse!

  31. Example • What’s the log27? • Get log(7) in your calculator ≈ 0.845098 • Get log(2) in your calculator ≈ 0.30103 • Log27 = Log(7)/Log(2) ≈ 0.845098/0.30103 • Log27≈2.80735

  32. Example problem • Find the domain of 2log7(4x-3)+7x-9 Whatever is inside the log has to be >0. I can find an answer whenever 4x-3>0 x>3/4

More Related